Gravitation in Flat Euclidean Spacetime Frame: Unified Electrogravity and Magnetogravity Forces
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An effective description of physics requires an appropriate geometrical frame. Threedimensional Euclidean space provides the geometrical frame for nonrelativistic physics. A derivation of an imaginary temporal axis −icˆq the speed, ˆq the unit wavevector of light, extends the standard Euclidean space into a welldefined fourdimensional Euclidean spacetime frame, which provides the natural mathematical framework for relativistic physics. The basic elements of the Euclidean spacetime frame are fully specified fourcomponent complex vectors satisfying standard vector operations and vector identities. In developing a theory of gravitation in the Euclidean spacetime frame, we have used the LenseThirring spacetime metric of linearized general relativity to derive an appropriate complex fourcomponent gravitational field potential vector. Taking the curl of the field potential vector provides a unified complex gravitational field strength composed of electrictype and magnetictype components. Taking the crossproduct of the complex fourcomponent velocity and the field strength provides a unified complex gravitational force intensity composed of gravitoelectric and gravitomagnetic components. Application to the motion of a gyroscope in the gravitational field of the earth provides the standard results of framedragging and geodetic effects as determined in linearized general relativity theory.
Introduction
So far, the fourdimensional Minkowski spacetime frame has provided the geometrical framework for describing the more accurate relativistic dynamics. But a serious, though unnoticed, drawback of the Minkowski spacetime frame is that it does not specify a unit vector in the temporal direction, yet all the three associated spatial coordinate axes, namely the xaxis, yaxis and zaxis are specified by their respective unit vectors i^, j^ and k^. This means that in the Minkowski spacetime frame, the fourcomponent vectors, i.e., fourvectors, can only be expressed in terms of their components in contravariant or covariant forms, but not in the standard form for representing the threecomponent vectors in the threedimensional Euclidean space frame. Mathematical operations with fourvectors in the Minkowski spacetime frame do not follow exactly the same procedure as the standard mathematical operations with the threecomponent vectors in the corresponding threedimensional Euclidean space. In particular products or partial derivatives such as divergence and curl of fourvectors are conveniently evaluated if expressed in contravariant and covariant form. Even though the dotproduct and the divergence of Minkowski fourvectors can be obtained in the standard Euclidean form if the temporal component is defined to be pure imaginary, the crossproduct and the curl of the fourvectors with real or pure imaginary temporal components cannot be obtained in the standard Euclidean form, but can only be evaluated in terms of their generalized components expressed in appropriate tensor form. This leads to the inescapable conclusion that the specification of the fourdimensional Minkowski geometrical framework with real or imaginary temporal axis as an extension of the threedimensional Euclidean space is not complete. This is essentially due to the fact that the Minkowski framework does not explicitly specify the temporal axis unit vector, even though the mathematical operations imply that the temporal axis is orthogonal to all three spatial coordinate axes.
In recent work [1], [2], the present author derived the unit vector along the temporal axis and identified it as the unit wave vector q^ in the direction of propagation of light. It emerges that the temporal axis is an imaginary axis specified by the unit wave vector of light. This has provided a fully specified fourdimensional spacetime geometrical frame with an imaginary temporal axis which can be interpreted as a mathematically consistent extension of standard threedimensional Euclidean space to a fourdimensional Euclidean spacetime frame. Four vectors with pure imaginary components and the related mathematical operations in the fourdimensional Euclidean spacetime frame now take exactly the same form as in the standard threedimensional Euclidean space. Details of derivations, fourvector definitions, mathematical operations and fourvector identities in the fourdimensional Euclidean spacetime frame (complex spacetime frame) are presented in [1], [2]. We observe that an imaginary time it (imaginary temporal coordinate ict) was first introduced in fourdimensional spacetime frame by Minkowski [3].
Euclidean Spacetime Frame
In the specification of the Euclidean spacetime frame, the temporal unit vector q^ is taken to be of general orientation relative to the mutually orthogonal spatial axes unit vectors i^, j^ and k^ according to the general orientation conditions: (1)i^⋅j^=0;j^⋅k^=0;k^⋅i^=0;q^⋅i^≠0;q^⋅j^≠0;q^⋅k^≠0
The Euclidean spacetime frame is characterized by an imaginary temporal axis specified by the unit wave vector q^ of light propagation. We shall call a general complex fourcomponent vector defined within the Euclidean spacetime frame, specified by all four spacetime unit vectors q^, i^, j^, k^, a Euclidean fourvector. We call the complex fourcomponent gradient vector defined within the Euclidean spacetime frame the Euclidean fourgradient. We denote a Euclidean fourvector by an uppercase letter symbol such as V while the usual threecomponent vector in standard threedimensional Euclidean space is denoted by a boldfaced letter symbol such as V or a conventional symbol with an overarrow, such as O→, where necessary. The Euclidean fourgradient is denoted by ∇ while the usual threecomponent gradient vector in threedimensional Euclidean space is denoted by an overarrow in the form ∇→.
A general Euclidean fourvector U, event fourvector X, displacement fourvector dX, velocity fourvector V and gradient fourvector ∇ are defined in the form: (2)U=−ic∅q^+U→; X=−ictq^+r→; dX=−icdtq^+dr→; V=−icq^+v→; ∇=ic∂∂tq^+∇→where c is the speed of light, r→ the position vector and v→ the velocity defined as usual in threedimensional Euclidean space. With Euclidean fourvectors taking the standard vector form with all unit vectors specified according to (2), mathematical operations within the fourdimensional Euclidean spacetime frame follow exactly the same procedure as the wellestablished mathematical operations with threecomponent vectors in the standard threedimensional Euclidean space.
The basic mathematical operations such as addition and subtraction, dot and cross products, divergence and curl of Euclidean fourvectors, and the Euclidean gradient of a scalar, are defined and evaluated in the standard vector operation forms: W=U±Q=−ic(Uq±Qq)q^+(U→±Q→); U⋅Q=U→⋅Q→−c2UqQq−icq^⋅(UqQ→+U→Qq) U×Q=U→×Q→−icq^×(UqQ→−U→Qq); ∇×Q=∇→×Q→−i(1c∂Q→∂t+∇→(cQq))×q^ (3)∇⋅Q=∂Qq∂t+∇→⋅Q→+iq^⋅(1c∂Q→∂t−∇→(cQq)); ∇∅=ic∂∅∂tq^+∇→∅
In detailed calculations presented in [1], [2], we have proved that Euclidean fourvectors satisfy exactly all forms of standard vector identities for threecomponent vectors in threedimensional Euclidean space. For completeness in defining the mathematical operations in the Euclidean spacetime frame, we present the final results of the basic Euclidean fourvector identities, which for a scalar function ∅ and Euclidean fourvectors U, Q, W have been obtained as: (4a)∇×∇∅=0; ∇⋅(∇×U)=0; ∇×(∇×U)=∇(∇⋅U)−∇2U (4b)∇⋅(∅U) =∇∅⋅U+∅∇⋅U; ∇×(∅∇)=∇∅×U+∅∇×U; ∇⋅(U×Q) =Q⋅(∇×U)−U⋅(∇×Q) (4c)∇(U⋅Q)=U×(∇×Q)+Q×(∇×U)+(U⋅∇)Q+(Q⋅∇)U (4d)∇×(U×Q)=U(∇⋅Q)−Q(∇⋅U)−(Q⋅∇)U−(U⋅∇)Q (4e)W×(U×Q)=U(W⋅Q)−Q(W⋅U);W×(U×Q)+U×(Q×W)+Q(W×U)=0 (4f)U⋅(U×Q)=0; U⋅(Q×W)=Q⋅(W×U)=W⋅(U×Q)
The basic algebraic operations presented in (4)–(10) constitute the mathematical foundation for formulating theories of dynamics of physical systems such as atoms in Euclidean spacetime frames.
Physics in Euclidean Spacetime Frame
In a systematic formulation of physics in the Euclidean spacetime frame, we apply the mathematical property that the basic elements of the Euclidean spacetime frame are Euclidean fourvectors, which as defined above, are complex fourcomponent vectors. The basic dynamical elements, the mechanical elements and the electromagnetic elements, of a physical system are defined as Euclidean fourvectors. As usual, the fundamental physical properties of matter are mass and electric charge, which generate appropriately defined force fields. We interpret a force field, with the generating mass or electric charge at the center (origin), as a bounded fourdimensional Euclidean spacetime frame. A mechanical (massgenerated) or electromagnetic (electric chargegenerated) force field is characterized by a Euclidean field potential fourvector. In general, mathematical operations with the physical Euclidean fourvectors have algebraic properties which reveal fundamental features of dynamics within a Euclidean spacetime frame.
Consistently with standard descriptions of physics in various geometrical frames, we identify the basic dynamical properties of physics in the Euclidean spacetime frame as mass m and electric charge q. These basic dynamical properties are defined as the temporal components of the respective Euclidean linear momentum and electric current density fourvectors P,J defined by (5)P=mV=−imcq^+p→; J=−icρq^+J→; p→=mv→; v→=dr→dtwhere v→ is the velocity, p→ linear momentum, ρ electric charge density and J→ electric current density, all defined as usual in threedimensional Euclidean space. The Euclidean linear momentum fourvector is equivalently interpreted as energymomentum fourvector defined by introducing relativistic energy E=mc2 in the form [4], [5]. (6)P=−iEcq^+p→; E=mc2
Other physical quantities such as orbital and spin angular momenta, together with the associated orbital and spin magnetic moments, etc., can be defined as desired.
In general, physics in the Euclidean spacetime frame naturally satisfies the basic conservation laws and invariance under Lorentz transformation. Application of the conservation laws and invariance under the Lorentz transformation easily provide the fundamental relativistic properties of time dilation and mass increase with speed, which have been obtained together with the corresponding spacetime metric ds and conserved energy E in the form [1], [2]. (7)dt=η−12γdτ; ds=η12γ−1cdt; m=η−12γm0; E2=p2c2+η−1m02c4where τ is the proper time, m0 the rest mass, γ the usual Lorentz transformation factor and η a Euclidean spacetime frame modification factor obtained in the form (8)γ=11−v2c2; η=1+4(q^.v→)2c2γ−2
In the special case where the temporal unit vector q^ is orthogonal to all the three spatial unit vectors i^,j^,k^, giving q^.p→=0, η=1, the fundamental relativistic properties in (7) reduce to the standard forms within the Minkowski spacetime frame associated with Einstein's special theory of relativity.
In the Euclidean spacetime frame in our case the gravitational field, a force field is characterized by a Euclidean field potential fourvector A and defined by (9)A=−ic∅q^+A→where c∅ is the scalar and A→ is the vector potential. The field strength F is obtained as the curl of the field potential fourvector A and the force intensity Fi is obtained as the crossproduct of the velocity fourvector V and the field strength in the form (10)F=∇×A; Fi=1cV×F
The Euclidean field potential fourvector, with the associated field strength and force intensity in (9), (10) are defined in general form, which can be applied to any force field such as electromagnetic and gravitational fields in a flat spacetime frame; only the dynamical properties of the field are specified in each case. We now develop an application to gravitation in a flat Euclidean spacetime frame in the next section.
The Euclidean Gravitational Field
An elegant property which has emerged is that the mathematical operations with Euclidean fourvectors provide a natural transition from nonrelativistic physics in threedimensional Euclidean space to relativistic physics in fourdimensional Euclidean spacetime frame. This property means that the Euclidean spacetime frame may be considered as the natural geometrical framework for developing the basic relativistic theory of gravitation, being a fundamental example of a general Euclidean mechanical force field.
We now introduce the Euclidean gravitational force field. To determine the appropriate form of the gravitational Euclidean field potential fourvector, we begin by observing that in Einstein’s general theory of relativity, the gravitational field potential is defined by a rank2 symmetric metric tensor in Riemann geometry. The general form of the metric tensor characterizes gravitation in a noninertial spacetime frame. Noting that in a weak gravitational field, linearization of Einstein’s general theory of relativity reduces to gravitoelectromagnetism (GEM), which governs relativistic gravitation in an inertial flat fourdimensional Minkowski spacetime frame [6]–[12], we use the standard LenseThirring spacetime metric in GEM to derive the appropriate form of the (weak) gravitational field potential fourvector. We developed the procedure for determining the GEM field potential fourvector within the flat Minkowski spacetime frame in [13], which we review briefly.
In GEM, the standard LenseThirring spacetime metric is obtained in the form [6]–[11], [13]. (11a)ds2=(1+2∅)dτ2+2h→⋅dr→dτ−(1−2∅)dr→2which we reorganize to bring the terms involving dτ together as (11b)ds2={(1+2∅)+2h→⋅v→}dτ2−(1−2∅)dr→2; v→=dr→dτwhere τ is the proper time and v→ is the velocity of a moving mass, such as a gyroscope in the GEM field. Introducing the speed of light c as appropriate in the temporal component of (11b) gives the form (11c)ds2={1c(1+2∅)c+2h→⋅v→}dτ2−(1−2∅)dr→2where we now recognize the GEM field scalar and vector components 1c(1+2∅), 2h→ together with the corresponding velocity fourvector components c, v→. This allows us to express the coefficient of the temporal component in (11c) in standard spacetime covariant form (11d)ds2=AgμVμdτ2−(1−2∅)dr→2; μ=0,1,2,3where we identify Agμ as the GEM field potential fourvector and Vμ as the velocity fourvector of the moving mass obtained in respective forms (11e)Agμ=(∅g,A→g); ∅g=1c(1+2∅); Ag=2h→; Vμ=(c,−v→)
We have thus used the LenseThirring spacetime metric to determine the appropriate form of the GEM field potential fourvector Agμ and the corresponding form of the velocity fourvector Vμ of a mass moving in the GEM field.
The GEM field potential fourvector Agμ=(∅g,A→g) determined from the LenseThirring spacetime metric of linearised general relativity characterizes a weak gravitational force field in a flat fourdimensional Minkowski spacetime frame. In the completely defined Euclidean spacetime frame where the imaginary temporal axis is specified by a unit vector, the GEM field potential Agμ is now defined as the Euclidean gravitational field potential fourvector Ag in the form (12)Ag=−ic∅gq^+A→g; ∅g=1c(1+2∅); A→g=2h→
With the Euclidean gravitational field potential fourvector determined as in (12), the Euclidean gravitational field strength, force and related dynamical quantities in relativistic gravitation can now be obtained.
The gravitational field strength F is obtained as the curl of the gravitational field potential fourvector Ag according to (10), which on applying the general form of the curl in (3), takes the form (13)F=B→g+iE→g×q^; B→g=∇→×A→g; E→g=−∇→(c∅g)−1c∂A→g∂twhere, by definition, the field strength component B→g generates deflection, while E→g generates translational motion in the Euclidean gravitational force field. Comparison with standard electromagnetic field strength leads to the interpretation that the component B→g generates magnetictype effects, while the component E→g generates electrictype effects in the gravitational force field. Consequently, we identify the component E→g as the gravitomagnetic field strength and the component E→g as the gravitoelectric field strength. In the nonrelativistic limit, the gravitoelectric field strength E→g reduces to the familiar Newton’s gravity field strength or gravitational acceleration g→. The important dynamical property emerging in (13) is that the Euclidean gravitation field strength F is a unified electromagnetictype field strength, which is generally called gravitoelectromagnetic (GEM) field strength in linearized general relativity theory [6]–[12].
The gravitational force intensity Fi in the Euclidean gravitational force field is obtained as the crossproduct of the Euclidean density fourvector V and the field strength F according to (10). Substituting V from (2) and F from (13) into the definition of Fi in (10), then applying the general form of the crossproduct from (3) provides the final form: (14)Fi=FL+iFO; FL=E→g+v→c×B→g−(q^.E→g)q^; FO=B→g×q^+v→c×(E→g×q^)
We identify FL as a Lorentztype (electrictype) force intensity and FO as an orbital magnetictype force intensity. The gravitational force intensity Fi is therefore a unified force composed of a Lorentz electrictype component FL which we define as a gravitoelectric force intensity and an orbital magnetictype component FO which we define as a gravitomagnetic force intensity.
Noting that B→g×q^ is normal to the plane between B→g and q^, and that E→g×q^ is normal to the plane between E→g and q^, we redefine the gravitomagnetic force intensity FO in the familiar form: (15)FO=B→g↑+v→c×E→g↑; B→g↑=B→g×q^; E→g↑=E→g×q^
We observe that, apart from a missing factor 32 in the second component, the gravitomagnetic force intensity obtained here in (18) or (19)agrees with the form obtained in the calculations of framedragging and geodetic effects using the geodesic equation for the spin fourvector of a gyroscope in linearized general relativity theory [7], [8]. We also notice that the Lorentztype force has a component that is induced by the motion and another term directed in the temporal axis as additional terms. Force is the rate of change of momentum, so we can write (16)dP→dt=mE→g+v→c×mB→g−m(q^.E→g)q^
The other component can be interpreted as the rate of change of spin (17)dmS→dt=mB→g×q^+v→c×(mE→g×q^)
It is the magnetictype forces that cause a change in the spin of a gyroscope moving in the gravitational field of the earth. Relativistic effects are therefore obtained by taking the crossproduct of the spinvector and the magnetictype forces. (18)gm ′2mcS→×(B→g×q^+v→c×(E→g×q^)); g=2; m ′=m
Finally, we have the framedragging and geodetic effects (19)Ω→FD=−32mB→g↑×S→; Ω→GE=−32v→c×mE→g↑×S→
Applications
From Astronomical point of view, reference frames serve as the observational perspectives from which we perceive the motion of objects in spacetime. When a massive object, such as a rotating black hole, drags spacetime along with its rotation, the effect is observable from different reference frames. From the perspective of an observer on the site to the rotating mass, the dragging of spacetime appears minimal or nonexistent. However, for an observer distant from the rotating mass, the framedragging effect becomes more pronounced, influencing the motion of nearby objects and altering their trajectories. Therefore, reference frames provide the framework for understanding relativistic effects that depend on the observer’s motion relative to the rotating mass [14].
Similarly, geodetic effects are intimately connected to reference frames in general relativity. These effects demonstrate the idea of curvature of spacetime caused by massive objects. It is the point to consider when explaining the fact that objects follow geodesic paths. The choice of reference frame affects how these geodesic paths are perceived. In a freely falling reference frame, where an observer experiences no gravitational forces, objects follow geodesic paths dictated by the curvature of spacetime. However, in a noninertial reference frame, such as one attached to a massive body like the Earth, the apparent motion of objects is influenced by additional forces, such as gravitational acceleration, which can obscure the underlying geodetic effects. Therefore, reference frames are essential for disentangling the effects of curvature from other forces and understanding the true nature of geodetic motion in curved spacetime. By studying these effects, we can validate and refine our understanding of general relativity, which serves as the cornerstone of modern theoretical physics [15].
In astrophysics, frame dragging influences the behaviour of massive rotating objects such as black holes and neutron stars. This effect affects the orbits of nearby objects and the emission of gravitational waves, providing valuable insights into the dynamics of these systems [16]. Observations of framedragging contribute to our understanding of astrophysical phenomena and help confirm the predictions of general relativity in extreme gravitational environments. For example, the detection of gravitational waves from merging black holes provides direct evidence of frame dragging effects and offers insights into the properties of these enigmatic objects [17].
Understanding relativistic effects is essential for precise navigation systems, particularly in space missions where accurate positioning is critical. Corrections based on general relativity, including geodetic effects, are necessary for achieving the high levels of accuracy required for GPS applications on Earth and in space. For instance, the Global Positioning System relies on corrections derived from general relativity to ensure accurate positioning and timing information for navigation purposes.
Conclusion
As it is, we have demonstrated that it is possible to handle gravitoelectromagnetism using vector mathematics. It is quite clear now that the motion of a body in the gravitational field induces an additional component to the component that was identified by Newton. The practical implications of relativistic effects extend beyond theoretical physics, influencing our understanding of celestial mechanics, the behaviour of compact astrophysical objects, and even the design of space missions. As we continue to push the boundaries of our understanding, these phenomena will undoubtedly remain at the forefront of gravitational research, guiding us toward new frontiers in both theoretical and observational exploration.
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